Geometry of Quantum Hall States: Gravitational Anomaly and Kinetic Coefficients

TL;DR

This paper links the universal transport coefficients of fractional quantum Hall states to geometric responses, especially gravitational anomalies, using new methods involving correlation functions and geometric invariants.

Contribution

It introduces a general framework to compute FQH correlation functions in curved space and relates transport properties to geometric invariants and gravitational anomalies.

Findings

Transport coefficients are governed by gravitational anomalies.

Developed two methods: Ward identity iteration and path integral formulation.

Established a connection between FQHE geometry and geometric invariants.

Abstract

We show that universal transport coefficients of the fractional quantum Hall effect (FQHE) can be understood as a response to variations of spatial geometry. Some transport properties are essentially governed by the gravitational anomaly. We develop a general method to compute correlation functions of FQH states in a curved space, where local transformation properties of these states are examined through local geometric variations. We introduce the notion of a generating functional and relate it to geometric invariant functionals recently studied in geometry. We develop two complementary methods to study the geometry of the FQHE. One method is based on iterating a Ward identity, while the other is based on a field theoretical formulation of the FQHE through a path integral formalism.